Isotropic local laws for sample covariance and generalized Wigner matrices
László Erdős (IST Austria)
Antti Knowles (ETH Zürich)
Horng-Tzer Yau (Harvard University)
Jun Yin (University of Wisconsin)
Abstract
We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v , (X^* X - z)^{-1}w\rangle - \langle v , w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v , w \in \mathbb{C}^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $\Im z \geq N^{-1+\varepsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-53
Publication Date: March 15, 2014
DOI: 10.1214/EJP.v19-3054
References
- Baik, Jinho; Ben Arous, Gérard; Péché, Sandrine. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 (2005), no. 5, 1643--1697. MR2165575
- A. Bloemendal, A. Knowles, H.-T. Yau, and J. Yin, Eigenvectors of deformed random matrices, In preparation.
- Bloemendal, Alex; Virág, Bálint. Limits of spiked random matrices I. Probab. Theory Related Fields 156 (2013), no. 3-4, 795--825. MR3078286
- Bloemendal, Alex; Virág, Bálint. Limits of spiked random matrices I. Probab. Theory Related Fields 156 (2013), no. 3-4, 795--825. MR3078286
- L. Erdös, Universality for random matrices and log-gases, Lecture Notes for Current Developments in Mathematics, Preprint arXiv:1212.0839 (2012).
- Erdős, László; Knowles, Antti; Yau, Horng-Tzer. Averaging fluctuations in resolvents of random band matrices. Ann. Henri Poincaré 14 (2013), no. 8, 1837--1926. MR3119922
- Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. Delocalization and diffusion profile for random band matrices. Comm. Math. Phys. 323 (2013), no. 1, 367--416. MR3085669
- Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. Spectral statistics of Erdős-Rényi Graphs II: Eigenvalue spacing and the extreme eigenvalues. Comm. Math. Phys. 314 (2012), no. 3, 587--640. MR2964770
- Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. The local semicircle law for a general class of random matrices. Electron. J. Probab. 18 (2013), no. 59, 58 pp. MR3068390
- Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. Spectral statistics of Erdős-Rényi graphs I: Local semicircle law. Ann. Probab. 41 (2013), no. 3B, 2279--2375. MR3098073
- Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 (2009), no. 2, 641--655. MR2481753
- Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Universality of random matrices and local relaxation flow. Invent. Math. 185 (2011), no. 1, 75--119. MR2810797
- Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer; Yin, Jun. The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 1, 1--46. MR2919197
- Erdős, László; Yau, Horng-Tzer; Yin, Jun. Universality for generalized Wigner matrices with Bernoulli distribution. J. Comb. 2 (2011), no. 1, 15--81. MR2847916
- Erdős, László; Yau, Horng-Tzer; Yin, Jun. Bulk universality for generalized Wigner matrices. Probab. Theory Related Fields 154 (2012), no. 1-2, 341--407. MR2981427
- Erdős, László; Yau, Horng-Tzer; Yin, Jun. Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229 (2012), no. 3, 1435--1515. MR2871147
- Johnstone, Iain M. On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2001), no. 2, 295--327. MR1863961
- Knowles, Antti; Yin, Jun. The isotropic semicircle law and deformation of Wigner matrices. Comm. Pure Appl. Math. 66 (2013), no. 11, 1663--1750. MR3103909
- Knowles, Antti; Yin, Jun. The outliers of a deformed wigner matrix, to appear in Ann. Prob. Preprint arXiv:1207.5619.
- V. A. Marchenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mat. Sbornik 72 (1967), 457--483.
- N. S. Pillai and J. Yin, Universality of covariance matrices, Preprint arXiv:1110.2501.
- Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 (1955), 548--564. MR0077805
This work is licensed under a Creative Commons Attribution 3.0 License.